Trad Cat is another name for Traditionalist Roman Catholics, especially those who are traditional to the extreme.
Trad cats generally attend Latin Masses, insist on their daughters wearing long skirts at all times, and don't very much like anything that is post-Vatican II.
They also tend to believe that sex is only to be had for baby making and must be done in the dark without any enjoyment.

Also known as "tradcat".

John's dad is such a trad cat.

You might not want to go to her house, her family is a bunch of trad cats.